DLMF:10.11.E4 (Q3106): Difference between revisions

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Property / Symbols used
 
Property / Symbols used: Q11426 / rank
 
Normal rank
Property / Symbols used: Q11426 / qualifier
 
Defining formula:

z 𝑧 {\displaystyle{\displaystyle z}}

z
Property / Symbols used: Q11426 / qualifier
 
xml-id: C10.S1.XMD6.m1cdec

Revision as of 15:20, 2 January 2020

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DLMF:10.11.E4
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    Statements

    Defining formula
    sin ⁡ ( ν ⁢ π ) ⁢ H ν ( 2 ) ⁡ ( z ⁢ e m ⁢ π ⁢ i ) = e ν ⁢ π ⁢ i ⁢ sin ⁡ ( m ⁢ ν ⁢ π ) ⁢ H ν ( 1 ) ⁡ ( z ) + sin ⁡ ( ( m + 1 ) ⁢ ν ⁢ π ) ⁢ H ν ( 2 ) ⁡ ( z ) . 𝜈 𝜋 Hankel-H-2-Bessel-third-kind 𝜈 𝑧 superscript 𝑒 𝑚 𝜋 𝑖 superscript 𝑒 𝜈 𝜋 𝑖 𝑚 𝜈 𝜋 Hankel-H-1-Bessel-third-kind 𝜈 𝑧 𝑚 1 𝜈 𝜋 Hankel-H-2-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle\sin\left(\nu\pi\right){H^{(2)}_{\nu}}\left(ze^{m% \pi i}\right)=e^{\nu\pi i}\sin\left(m\nu\pi\right){H^{(1)}_{\nu}}\left(z\right% )+\sin\left((m+1)\nu\pi\right){H^{(2)}_{\nu}}\left(z\right).}}
    0 references
    DLMF id
    DLMF:10.11.E4
    0 references
    Symbols used
    Bessel function of the third kind (or Hankel function)
    Defining formula
    H ν ( 1 ) ⁡ ( z ) Hankel-H-1-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle{H^{(1)}_{\NVar{\nu}}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E5.m2aadec
    0 references
    Bessel function of the third kind (or Hankel function)
    Defining formula
    H ν ( 2 ) ⁡ ( z ) Hankel-H-2-Bessel-third-kind 𝜈 𝑧 {\displaystyle{\displaystyle{H^{(2)}_{\NVar{\nu}}}\left(\NVar{z}\right)}}
    xml-id
    C10.S2.E6.m2aadec
    0 references
    the ratio of the circumference of a circle to its diameter
    Defining formula
    π {\displaystyle{\displaystyle\pi}}
    xml-id
    C3.S12.E1.m2acdec
    0 references
    base of natural logarithm
    Defining formula
    e {\displaystyle{\displaystyle\mathrm{e}}}
    xml-id
    C4.S2.E11.m2acdec
    0 references
    imaginary unit
    Defining formula
    i imaginary-unit {\displaystyle{\displaystyle\mathrm{i}}}
    xml-id
    C1.S9.E1.m2acdec
    0 references
    sine function
    Defining formula
    sin ⁡ z 𝑧 {\displaystyle{\displaystyle\sin\NVar{z}}}
    xml-id
    C4.S14.E1.m2abdec
    0 references
    Q11438
    Defining formula
    m 𝑚 {\displaystyle{\displaystyle m}}
    xml-id
    C10.S1.XMD1.m1cdec
    0 references
    Q11426
    Defining formula
    z 𝑧 {\displaystyle{\displaystyle z}}
    xml-id
    C10.S1.XMD6.m1cdec
    0 references
     
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